In traditional first-order predicate calculus a ``term'' is a
syntactic entity denoting some object in the universe of
individuals. Often, for example, the syntactic characterization of
a term is that it is either a variable symbol or the application of
a function symbol to the appropriate number of argument terms.
Traditionally, ``atomic formulas'' are built from terms with
predicate symbols such as ``equal'' and ``member;'' ``formulas'' are
then built from atomic formulas with propositional ``operators''
like ``not,'' ``and,'' and ``implies.'' Theorems are formulas.
Theorems are ``valid'' in the sense that the value of a theorem is
true, in any model of the axioms and under all possible assignments
of individuals to variables.

However, in ACL2, terms are used in place of both atomic formulas
and formulas. ACL2 does not have predicate symbols or propositional
operators as distinguished syntactic entities. The ACL2 universe of
individuals includes a ``true'' object (denoted by t) and a
``false'' object (denoted by nil), predicates and propositional
operators are functions that return these objects. Theorems in ACL2
are terms and the ``validity'' of a term means that, under no
assignment to the variables does the term evaluate to nil.

We use the word ``term'' in ACL2 in three distinct senses. We will
speak of ``translated'' terms, ``untranslated'' terms, and
``pseudo-'' terms.

Translated Terms: The Strict Sense and Internal Form

In its most strict sense, a ``term'' is either a legal variable
symbol, a quoted constant, or the application of an n-ary function
symbol or closed lambda expression to a true list of n terms.

The legal variable symbols are symbols other than t or nil
which are not in the keyword package, do not start with ampersand,
do not start and end with asterisks, and if in the main Lisp
package, do not violate an appropriate restriction (see name).

Quoted constants are expressions of the form (quote x), where x is
any ACL2 object. Such expressions may also be written 'x.

Closed lambda expressions are expressions of the form
(lambda (v1 ... vn) body) where the vi are distinct legal
variable symbols, body is a term, and the only free variables in
body are among the vi.

The function termp, which takes two arguments, an alleged term x and
a logical world w (see world), recognizes terms of a given
extension of the logic. Termp is defined in :program mode.
Its definition may be inspected with :petermp for a complete
specification of what we mean by ``term'' in the most strict sense.
Most ACL2 term-processing functions deal with terms in this strict
sense and use termp as a guard. That is, the ``internal form''
of a term satisfies termp, the strict sense of the word ``term.''

Untranslated Terms: What the User Types

While terms in the strict sense are easy to explore (because their
structure is so regular and simple) they can be cumbersome to type.
Thus, ACL2 supports a more sugary syntax that includes uses of
macros and constant symbols. Very roughly speaking, macros are
functions that produce terms as their results. Constants are
symbols that are associated with quoted objects. Terms in this
sugary syntax are ``translated'' to terms in the strict sense; the
sugary syntax is more often called ``untranslated.'' Roughly
speaking, translation just implements macroexpansion, the
replacement of constant symbols by their quoted values, and the
checking of all the rules governing the strict sense of ``term.''

More precisely, macro symbols are as described in the documentation
for defmacro. A macro, mac, can be thought of as a function,
mac-fn, from ACL2 objects to an ACL2 object to be treated as an
untranslated term. For example, caar is defined as a macro symbol;
the associated macro function maps the object x into the object
(car (car x)). A macro form is a ``call'' of a macro symbol,
i.e., a list whose car is the macro symbol and whose cdr is an
arbitrary true list of objects, used as a term. Macroexpansion is
the process of replacing in an untranslated term every occurrence of
a macro form by the result of applying the macro function to the
appropriate arguments. The ``appropriate'' arguments are determined
by the exact form of the definition of the macro; macros support
positional, keyword, optional and other kinds of arguments.
See defmacro.

In addition to macroexpansion and constant symbol dereferencing,
translation implements the mapping of let and let* forms into
applications of lambda expressions and closes lambda expressions
containing free variables. Thus, the translation of

(let ((x (1+ i))) (cons x k))

can be seen as a two-step process that first produces

((lambda (x) (cons x k)) (1+ i))

and then

((lambda (x k) (cons x k)) (1+ i) k) .

Observe that the body of the let and of the first lambda
expression contains a free k which is finally bound and passed
into the second lambda expression.

When we say, of an event-level function such as defun or defthm,
that some argument ``must be a term'' we mean an untranslated term.
The event functions translate their term-like arguments.

To better understand the mapping between untranslated terms and
translated terms it is convenient to use the keyword command :trans
to see examples of translations. See trans and also
see trans1.

Finally, we note that the theorem prover prints terms in
untranslated form. But there can be more than one correct untranslated
term corresponding to a given translated term. For example, the
translated term (if x y 'nil) can be untranslated as (if x y nil)
and can also be untranslated as (and x y). The theorem prover
attempts to print an untranslated term that is as helpful to the
user as possible. In particular, consider a term of the form
(nth k st) where st is a single-threaded object
(see stobj) and the kth accessor of st is, say, kn. The
theorem prover typically would expand (kn st) to (nth k st). If
k is large then it could be difficult for the user to make sense out
of a proof transcript that mentions the expanded term. Fortunately,
the untranslation of (nth k st) would be (nth *kn* st); here
*kn* would be a constant (see defconst) added by the defstobj
event introducing st, defined to have value k. The user can
extend this user-friendly style of printing applications of nth to
stobjs; see add-nth-alias. These remarks about printing
applications of function nth extend naturally to function
update-nth. Moreover, the prover will attempt to treat terms as
stobjs for the above purpose when appropriate. For example, if
function foo has signature((foo * st) => (mv * * * st)), where
st is introduced with (defstobj st f0 f1), then the term(nth '1 (mv-nth '3 (foo x st0))) will be printed as
(nth *f1* (mv-nth 3 (foo x st0))).

Pseudo-Terms: A Common Guard for Metafunctions

Because termp is defined in :program mode, it cannot be used
effectively in conjectures to be proved. Furthermore, from the
perspective of merely guarding a term processing function, termp
often checks more than is required. Finally, because termp
requires the logical world as one of its arguments it is impossible
to use termp as a guard in places where the logical world is not
itself one of the arguments.

For these reasons we support the idea of ``pseudo-terms.'' A
pseudo-term is either a symbol (but not necessarily one having the
syntax of a legal variable symbol), a true list beginning with quote
(but not necessarily well-formed), or the ``application of'' a
symbol or pseudo lambda expression to a true list of
pseudo-terms. A pseudo lambda expression is an expression of the
form (lambda (v1 ... vn) body) where the vi are all symbols
and body is a pseudo-term.

Pseudo-terms are recognized by the unary function pseudo-termp. If
(termp x w) is true, then (pseudo-termp x) is true. However, if x
fails to be a (strict) term it may nevertheless still be a
pseudo-term. For example, (car a b) is not a term, because car is
applied to the wrong number of arguments, but it is a pseudo-term.

The structures recognized by pseudo-termp can be recursively
explored with the same simplicity that terms can be. In particular,
if x is not a variablep or an fquotep, then (ffn-symb x) is the
function (symbol or lambda expression) and (fargs x) is the list of
argument pseudo-terms. A metafunction may use pseudo-termp as the
guard.